Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 6·13-s + 15-s − 2·17-s − 8·19-s + 8·23-s + 25-s + 27-s + 2·29-s − 4·31-s + 2·37-s − 6·39-s + 6·41-s − 4·43-s + 45-s − 8·47-s − 2·51-s − 10·53-s − 8·57-s + 4·59-s − 2·61-s − 6·65-s − 4·67-s + 8·69-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 0.280·51-s − 1.37·53-s − 1.05·57-s + 0.520·59-s − 0.256·61-s − 0.744·65-s − 0.488·67-s + 0.963·69-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(47040\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{47040} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 47040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.032756594$
$L(\frac12)$  $\approx$  $2.032756594$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.70718311756801, −14.24730955726925, −13.47991378519991, −12.99079080671071, −12.71192260134020, −12.22993971638427, −11.35218084537885, −10.98297513527322, −10.32067558484501, −9.906746109707224, −9.264976391969128, −8.908676915254073, −8.369401570558491, −7.624942831429000, −7.216054144267384, −6.564856683731167, −6.159602102024275, −5.197923831999591, −4.726313740864635, −4.331179485630834, −3.349310463175362, −2.764156666495984, −2.192853096675773, −1.627898078241162, −0.4607531177656057, 0.4607531177656057, 1.627898078241162, 2.192853096675773, 2.764156666495984, 3.349310463175362, 4.331179485630834, 4.726313740864635, 5.197923831999591, 6.159602102024275, 6.564856683731167, 7.216054144267384, 7.624942831429000, 8.369401570558491, 8.908676915254073, 9.264976391969128, 9.906746109707224, 10.32067558484501, 10.98297513527322, 11.35218084537885, 12.22993971638427, 12.71192260134020, 12.99079080671071, 13.47991378519991, 14.24730955726925, 14.70718311756801

Graph of the $Z$-function along the critical line